Deriving first integrals of differential equations and vice-versa

Question

Suppose I am given the system of ODE \begin{align} \dot x = & y\\ \dot y = & -V'(x) \, y \end{align} Obviously the function $F(x,y) = y + V(x)$ is a first integral of the system. Am I allowed to derive the following system? \begin{align} \dot x = & \frac{\partial F}{\partial y} = 1\\ \dot y = & - \frac{\partial F}{\partial x} = -V'(x) \end{align} Are the two systems equivalent? Obviosuly trajectories are the level set of $F(x,y)$ for both systems. Is it just a matter of parametrization? Thanks in advance.

Answer 1

No, they are not equivalent, but have the same solution curves as long as $y\ne 0$.

It is in general true that $\dot x=f(x)$ and $\dot x=f(x)\phi(x)$ have the same solution curves but different parametrizations on the set $\{\phi(x)\ne 0\}$.

If $x(t)$ solves $\dot x=f(x)$ and $\tau(s)$ is a monotonous differentiable function, then $$ \frac d{ds}x(τ(s))=f(x(τ(s)))τ'(s) $$ so that when $τ'(s)=ϕ(x(τ(s)))$ then the reparametrization $z(s)=x(τ(s))$ solves $\dot z=f(z)ϕ(z)$ without leaving the geometric curve traces by $x(t)$.